Question

If $z\ne 1$ and $\frac{z^2}{z-1}$ is real, then the point represented by the complex numbe $z$ lies

• A. either on the real axis or on a circle passing through the origin
• B. on a circle with centre at the origin
• C. either on the real axis or on a circle not passing through the origin
• D. on the imaginary axis

Answer 1

The correct option is A either on the real axis or on a circle passing through the origin

Let $z=x+iy(\because x\ne 1\text{as}z\ne 1)$
$z^2=(x^2-y^2)+i\left(2xy\right)$
Imaginary part of $\frac{z^2}{z-1}=0$
$\Rightarrow 2xy(x-1)-y(x^2-y^2)=0$
$\Rightarrow y=0;x^2+y^2-2x=0$
$\therefore z$ lies either on real axis or on a circle passing through origin.